Supplemental

A Comparison to Latent Growth Curve Models

Latent growth curve (LGC) models are in a sense, just a different form of the very commonly used mixed model framework. In some ways they are more flexible, mostly in the standard structural equation modeling framework that allows for indirect, and other complex covariate relationships. In other ways, they are less flexible, e.g. requiring balanced data, estimating nonlinear relationships, data with many time points, dealing with time-varying covariates. With appropriate tools there is little one can’t do with the normal mixed model approach relative to the SEM approach, and one would likely have easier interpretation. As such I’d recommend sticking with the standard mixed model framework unless you really need to.

To best understand a growth curve model, I still think it’s instructive to see it from the mixed model perspective, where things are mostly interpretable from what you know from a standard linear model. We will use our GPA example from before.

Random Effects as Latent Varaibles

As before we assume the following for the student effects.

\[\mathcal{GPA} = (b_{\mathrm{intercept}} + \mathrm{intercept}_{\mathrm{student}}) + (b_{\mathrm{occ}} + \mathrm{occ}_{\mathrm{student}})\cdot \mathrm{occasion} + \epsilon\]

\[ \mathrm{intercept}_{\mathrm{student}} \sim \mathscr{N}(0, \tau)\] \[\mathrm{occ}_{\mathrm{student}} \sim \mathscr{N}(0, \varphi)\]

Thus the student effects are random, and specifically are normally distributed with mean of zero and some estimated standard deviation (\(\tau\), \(\varphi\) respectively). We consider these as unspecified, or latent, effects due to student.

Random Effects in SEM

In SEM, the latent variables are assumed normally distributed, usually with zero mean, and some estimated variance, just like the random effects in mixed models. Through this that we can maybe start to get a sense of random effects as latent variables (or vice versa). Indeed, mixed models have ties to many other kinds of models (e.g. spatial, additive), because they too add a ‘random’ component to the model in some fashion.

Running a Growth Curve Model

For those familiar with structural equation modeling (SEM), growth curve models will actually look a bit different compared to with typical SEM, because we have to fix the factor loadings to specific values in order to make it work. This also leads to non-standard output relative to other SEM models, as there is nothing to estimate for the many fixed parameters.

More specifically, we’ll have a latent variable representing the random intercepts, as well as one representing the random slopes. The visualization looks like a factor analysis, with a factor we are calling the intercepts and a factor we’re calling the slopes. Unlike with factor analysis, all loadings for the intercept factor are 1[^likematrix]. The loadings for the effect of time are arbitrary, but should accurately reflect the time spacing, and typically it is good to start at zero, so that the zero has a meaningful interpretation.

Wide Data

As might be guessed from the above visualization, for the LGC our data needs to be in wide format, where each row represents a person and we have separate columns for each time point of the target variable, as opposed to the long format we used in the previous mixed model. We can use the spread function from tidyr to help with that.

load('data/gpa.RData')
gpa_wide = gpa %>% 
  select(student, sex, highgpa, occasion, gpa) %>% 
  spread(key = occasion, value = gpa) %>% 
  rename_at(vars(`0`,`1`,`2`,`3`,`4`,`5`), function(x) glue::glue('semester_{x}'))

head(gpa_wide)
  student    sex highgpa semester_0 semester_1 semester_2 semester_3 semester_4 semester_5
1       1 female     2.8        2.3        2.1        3.0        3.0        3.0        3.3
2       2   male     2.5        2.2        2.5        2.6        2.6        3.0        2.8
3       3 female     2.5        2.4        2.9        3.0        2.8        3.3        3.4
4       4   male     3.8        2.5        2.7        2.4        2.7        2.9        2.7
5       5   male     3.1        2.8        2.8        2.8        3.0        2.9        3.1
6       6 female     2.9        2.5        2.4        2.4        2.3        2.7        2.8

We’ll use lavaan for our excursion into LGC. The syntax will require its own modeling code, but lavaan tries to keep to R regression model style. The names of intercept and slope are arbitrary. The =~ is just denoting that the left-hand side is the latent variable, and the right-hand side are the observed/manifest variables.

lgc_init_model = '
  intercept =~ 1*semester_0 + 1*semester_1 + 1*semester_2 + 1*semester_3 + 1*semester_4 + 1*semester_5
  slope =~ 0*semester_0 + 1*semester_1 + 2*semester_2 + 3*semester_3 + 4*semester_4 + 5*semester_5
'

Now we’re ready to run the model. Note that lavaan has a specific function, growth, to use for these models. It doesn’t spare us any effort for the model syntax, but does make it unnecessary to set various arguments for the more generic sem and lavaan functions.

library(lavaan)
lgc_init = growth(lgc_init_model, data = gpa_wide)
summary(lgc_init)
lavaan 0.6-3 ended normally after 73 iterations

  Optimization method                           NLMINB
  Number of free parameters                         11

  Number of observations                           200

  Estimator                                         ML
  Model Fit Test Statistic                      43.945
  Degrees of freedom                                16
  P-value (Chi-square)                           0.000

Parameter Estimates:

  Information                                 Expected
  Information saturated (h1) model          Structured
  Standard Errors                             Standard

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)
  intercept =~                                        
    semester_0        1.000                           
    semester_1        1.000                           
    semester_2        1.000                           
    semester_3        1.000                           
    semester_4        1.000                           
    semester_5        1.000                           
  slope =~                                            
    semester_0        0.000                           
    semester_1        1.000                           
    semester_2        2.000                           
    semester_3        3.000                           
    semester_4        4.000                           
    semester_5        5.000                           

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)
  intercept ~~                                        
    slope             0.002    0.002    1.629    0.103

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)
   .semester_0        0.000                           
   .semester_1        0.000                           
   .semester_2        0.000                           
   .semester_3        0.000                           
   .semester_4        0.000                           
   .semester_5        0.000                           
    intercept         2.598    0.018  141.956    0.000
    slope             0.106    0.005   20.338    0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .semester_0        0.080    0.010    8.136    0.000
   .semester_1        0.071    0.008    8.799    0.000
   .semester_2        0.054    0.006    9.039    0.000
   .semester_3        0.029    0.003    8.523    0.000
   .semester_4        0.015    0.002    5.986    0.000
   .semester_5        0.016    0.003    4.617    0.000
    intercept         0.035    0.007    4.947    0.000
    slope             0.003    0.001    5.645    0.000

Most of the output is blank, which is needless clutter, but we do get the same five parameter values we are interested in though.

Start with the ‘intercepts’:

Intercepts:
                   Estimate  Std.Err  Z-value  P(>|z|)

    intercept         2.598    0.018  141.956    0.000
    slope             0.106    0.005   20.338    0.000

It might be odd to call your fixed effects ‘intercepts’, but it makes sense if we are thinking of it as a multilevel model as depicted previously, where we actually broke out the random effects as a separate model. The estimates here are pretty much spot on with our mixed model estimates.

library(lme4)
gpa_mixed = lmer(gpa ~ occasion + (1 + occasion | student), data=gpa)
summary(gpa_mixed)
Linear mixed model fit by REML ['lmerMod']
Formula: gpa ~ occasion + (1 + occasion | student)
   Data: gpa

REML criterion at convergence: 261

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.2695 -0.5377 -0.0128  0.5326  3.1939 

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 student  (Intercept) 0.045193 0.21259       
          occasion    0.004504 0.06711  -0.10
 Residual             0.042388 0.20588       
Number of obs: 1200, groups:  student, 200

Fixed effects:
            Estimate Std. Error t value
(Intercept) 2.599214   0.018357  141.59
occasion    0.106314   0.005885   18.07

Correlation of Fixed Effects:
         (Intr)
occasion -0.345
# fixef(gpa_mixed)

Now let’s look at the variance estimates. The estimation of residual variance for each time point in the LGC distinguishes the two approaches, but not necessarily so. We could fix them to be identical here, or conversely allow them to be estimated in the mixed model framework. Just know that’s why the results are not identical (to go along with their respective estimation approaches, which are also different by default).

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)
  intercept ~~                                        
    slope             0.002    0.002    1.629    0.103
    
Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .semester_0        0.080    0.010    8.136    0.000
   .semester_1        0.071    0.008    8.799    0.000
   .semester_2        0.054    0.006    9.039    0.000
   .semester_3        0.029    0.003    8.523    0.000
   .semester_4        0.015    0.002    5.986    0.000
   .semester_5        0.016    0.003    4.617    0.000
    intercept         0.035    0.007    4.947    0.000
    slope             0.003    0.001    5.645    0.000
VarCorr(gpa_mixed)
 Groups   Name        Std.Dev. Corr  
 student  (Intercept) 0.212587       
          occasion    0.067111 -0.098
 Residual             0.205883       

The differences provide some insight. LGC by default assumes heterogeneous variance for each time point. Mixed models by default assume the same variance for each time point, but can allow them to be estimated separately in most modeling packages.

As an example, if we fix the variances to be equal, the models are now identical.

model = "
  intercept =~ 1*semester_0 + 1*semester_1 + 1*semester_2 + 1*semester_3 + 1*semester_4 + 1*semester_5
  slope =~ 0*semester_0 + 1*semester_1 + 2*semester_2 + 3*semester_3 + 4*semester_4 + 5*semester_5
  
  semester_0 ~~ residual*semester_0
  semester_1 ~~ residual*semester_1
  semester_2 ~~ residual*semester_2
  semester_3 ~~ residual*semester_3
  semester_4 ~~ residual*semester_4
  semester_5 ~~ residual*semester_5
"

growthCurveModel = growth(model, data=gpa_wide)
summary(growthCurveModel)
lavaan 0.6-3 ended normally after 51 iterations

  Optimization method                           NLMINB
  Number of free parameters                         11
  Number of equality constraints                     5

  Number of observations                           200

  Estimator                                         ML
  Model Fit Test Statistic                     191.409
  Degrees of freedom                                21
  P-value (Chi-square)                           0.000

Parameter Estimates:

  Information                                 Expected
  Information saturated (h1) model          Structured
  Standard Errors                             Standard

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)
  intercept =~                                        
    semester_0        1.000                           
    semester_1        1.000                           
    semester_2        1.000                           
    semester_3        1.000                           
    semester_4        1.000                           
    semester_5        1.000                           
  slope =~                                            
    semester_0        0.000                           
    semester_1        1.000                           
    semester_2        2.000                           
    semester_3        3.000                           
    semester_4        4.000                           
    semester_5        5.000                           

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)
  intercept ~~                                        
    slope            -0.001    0.002   -0.834    0.404

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)
   .semester_0        0.000                           
   .semester_1        0.000                           
   .semester_2        0.000                           
   .semester_3        0.000                           
   .semester_4        0.000                           
   .semester_5        0.000                           
    intercept         2.599    0.018  141.947    0.000
    slope             0.106    0.006   18.111    0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .smstr_0 (rsdl)    0.042    0.002   20.000    0.000
   .smstr_1 (rsdl)    0.042    0.002   20.000    0.000
   .smstr_2 (rsdl)    0.042    0.002   20.000    0.000
   .smstr_3 (rsdl)    0.042    0.002   20.000    0.000
   .smstr_4 (rsdl)    0.042    0.002   20.000    0.000
   .smstr_5 (rsdl)    0.042    0.002   20.000    0.000
    intrcpt           0.045    0.007    6.599    0.000
    slope             0.004    0.001    6.387    0.000

Compare to the lme4 output.

 Groups   Name        Variance  Corr  
 student  (Intercept) 0.0451934       
          occasion    0.0045039 -0.098
 Residual             0.0423879       

Random Intercepts

How can we put these models on the same footing? Let’s take a step back and do a model with only random intercepts. In this case, time is an observed measure, and has no person-specific variability. Our graphical model now looks like the following.

We can do this by fixing the slope ‘factor’ to have zero variance. However, note also that in the LGC, at each time point of the gpa outcome, we have a unique (residual) variance associated with it. Conversely, this is constant in the mixed model setting, i.e. we only have one estimate for the residual variance that does not vary by occasion. We deal with this in the LGC by giving the parameter a name and then applying it to each time point.

lgc_ran_int_model = '

 intercept =~ 1*semester_0 + 1*semester_1 + 1*semester_2 + 1*semester_3 + 1*semester_4 + 1*semester_5
 slope =~ 0*semester_0 + 1*semester_1 + 2*semester_2 + 3*semester_3 + 4*semester_4 + 5*semester_5
 
 slope ~~ 0*slope                  # slope variance is zero
 intercept ~~ 0*slope              # no covariance
 
 
 semester_0 ~~ resid*semester_0    # same residual variance for each time point
 semester_1 ~~ resid*semester_1
 semester_2 ~~ resid*semester_2
 semester_3 ~~ resid*semester_3
 semester_4 ~~ resid*semester_4
 semester_5 ~~ resid*semester_5
'

Now each time point will have one variance estimate. Let’s run the LGC.

lgc_ran_int = growth(lgc_ran_int_model, data = gpa_wide)
summary(lgc_ran_int, nd=4)  # increase the number of digits shown
lavaan 0.6-3 ended normally after 36 iterations

  Optimization method                           NLMINB
  Number of free parameters                          9
  Number of equality constraints                     5

  Number of observations                           200

  Estimator                                         ML
  Model Fit Test Statistic                     338.824
  Degrees of freedom                                23
  P-value (Chi-square)                           0.000

Parameter Estimates:

  Information                                 Expected
  Information saturated (h1) model          Structured
  Standard Errors                             Standard

Latent Variables:
                    Estimate   Std.Err   z-value   P(>|z|)
  intercept =~                                            
    semester_0        1.0000                              
    semester_1        1.0000                              
    semester_2        1.0000                              
    semester_3        1.0000                              
    semester_4        1.0000                              
    semester_5        1.0000                              
  slope =~                                                
    semester_0        0.0000                              
    semester_1        1.0000                              
    semester_2        2.0000                              
    semester_3        3.0000                              
    semester_4        4.0000                              
    semester_5        5.0000                              

Covariances:
                    Estimate   Std.Err   z-value   P(>|z|)
  intercept ~~                                            
    slope             0.0000                              

Intercepts:
                    Estimate   Std.Err   z-value   P(>|z|)
   .semester_0        0.0000                              
   .semester_1        0.0000                              
   .semester_2        0.0000                              
   .semester_3        0.0000                              
   .semester_4        0.0000                              
   .semester_5        0.0000                              
    intercept         2.5992    0.0217  120.0471    0.0000
    slope             0.1063    0.0041   26.1094    0.0000

Variances:
                    Estimate   Std.Err   z-value   P(>|z|)
    slope             0.0000                              
   .smstr_0 (resd)    0.0580    0.0026   22.3607    0.0000
   .smstr_1 (resd)    0.0580    0.0026   22.3607    0.0000
   .smstr_2 (resd)    0.0580    0.0026   22.3607    0.0000
   .smstr_3 (resd)    0.0580    0.0026   22.3607    0.0000
   .smstr_4 (resd)    0.0580    0.0026   22.3607    0.0000
   .smstr_5 (resd)    0.0580    0.0026   22.3607    0.0000
    intrcpt           0.0634    0.0073    8.6605    0.0000

Compare it to the corresponding mixed model.

summary(lme4::lmer(gpa ~ occasion + (1|student), data=gpa))
Linear mixed model fit by REML ['lmerMod']
Formula: gpa ~ occasion + (1 | student)
   Data: gpa

REML criterion at convergence: 408.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.6169 -0.6373 -0.0004  0.6361  2.8310 

Random effects:
 Groups   Name        Variance Std.Dev.
 student  (Intercept) 0.06372  0.2524  
 Residual             0.05809  0.2410  
Number of obs: 1200, groups:  student, 200

Fixed effects:
            Estimate Std. Error t value
(Intercept) 2.599214   0.021696   119.8
occasion    0.106314   0.004074    26.1

Correlation of Fixed Effects:
         (Intr)
occasion -0.469

Now we have identical results. Now let’s let the slope for occasion vary. We can just delete or comment out the syntax related to the (co-) variance. We will keep the variance constant.

lgc_ran_int_ran_slope_model = '

 intercept =~ 1*semester_0 + 1*semester_1 + 1*semester_2 + 1*semester_3 + 1*semester_4 + 1*semester_5
 slope =~ 0*semester_0 + 1*semester_1 + 2*semester_2 + 3*semester_3 + 4*semester_4 + 5*semester_5
 
 # slope ~~ 0*slope                  # slope variance is zero
 # intercept ~~ 0*slope              # no covariance
 
 
 semester_0 ~~ resid*semester_0    # same residual variance for each time point
 semester_1 ~~ resid*semester_1
 semester_2 ~~ resid*semester_2
 semester_3 ~~ resid*semester_3
 semester_4 ~~ resid*semester_4
 semester_5 ~~ resid*semester_5
'
lgc_ran_int_ran_slope = growth(lgc_ran_int_ran_slope_model, data = gpa_wide)
summary(lgc_ran_int_ran_slope, nd=4)  # increase the number of digits shown
lavaan 0.6-3 ended normally after 51 iterations

  Optimization method                           NLMINB
  Number of free parameters                         11
  Number of equality constraints                     5

  Number of observations                           200

  Estimator                                         ML
  Model Fit Test Statistic                     191.409
  Degrees of freedom                                21
  P-value (Chi-square)                           0.000

Parameter Estimates:

  Information                                 Expected
  Information saturated (h1) model          Structured
  Standard Errors                             Standard

Latent Variables:
                    Estimate   Std.Err   z-value   P(>|z|)
  intercept =~                                            
    semester_0        1.0000                              
    semester_1        1.0000                              
    semester_2        1.0000                              
    semester_3        1.0000                              
    semester_4        1.0000                              
    semester_5        1.0000                              
  slope =~                                                
    semester_0        0.0000                              
    semester_1        1.0000                              
    semester_2        2.0000                              
    semester_3        3.0000                              
    semester_4        4.0000                              
    semester_5        5.0000                              

Covariances:
                    Estimate   Std.Err   z-value   P(>|z|)
  intercept ~~                                            
    slope            -0.0014    0.0016   -0.8337    0.4045

Intercepts:
                    Estimate   Std.Err   z-value   P(>|z|)
   .semester_0        0.0000                              
   .semester_1        0.0000                              
   .semester_2        0.0000                              
   .semester_3        0.0000                              
   .semester_4        0.0000                              
   .semester_5        0.0000                              
    intercept         2.5992    0.0183  141.9471    0.0000
    slope             0.1063    0.0059   18.1113    0.0000

Variances:
                    Estimate   Std.Err   z-value   P(>|z|)
   .smstr_0 (resd)    0.0424    0.0021   20.0000    0.0000
   .smstr_1 (resd)    0.0424    0.0021   20.0000    0.0000
   .smstr_2 (resd)    0.0424    0.0021   20.0000    0.0000
   .smstr_3 (resd)    0.0424    0.0021   20.0000    0.0000
   .smstr_4 (resd)    0.0424    0.0021   20.0000    0.0000
   .smstr_5 (resd)    0.0424    0.0021   20.0000    0.0000
    intrcpt           0.0449    0.0068    6.5992    0.0000
    slope             0.0045    0.0007    6.3874    0.0000
summary(lme4::lmer(gpa ~ occasion + (1 + occasion|student), data=gpa))
Linear mixed model fit by REML ['lmerMod']
Formula: gpa ~ occasion + (1 + occasion | student)
   Data: gpa

REML criterion at convergence: 261

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.2695 -0.5377 -0.0128  0.5326  3.1939 

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 student  (Intercept) 0.045193 0.21259       
          occasion    0.004504 0.06711  -0.10
 Residual             0.042388 0.20588       
Number of obs: 1200, groups:  student, 200

Fixed effects:
            Estimate Std. Error t value
(Intercept) 2.599214   0.018357  141.59
occasion    0.106314   0.005885   18.07

Correlation of Fixed Effects:
         (Intr)
occasion -0.345

Note that the intercept-slope relationship in the LGC is expressed as a covariance. If we want correlation, we just ask for standardized output.

summary(lgc_ran_int_ran_slope, nd=4, std=T)
lavaan 0.6-3 ended normally after 51 iterations

  Optimization method                           NLMINB
  Number of free parameters                         11
  Number of equality constraints                     5

  Number of observations                           200

  Estimator                                         ML
  Model Fit Test Statistic                     191.409
  Degrees of freedom                                21
  P-value (Chi-square)                           0.000

Parameter Estimates:

  Information                                 Expected
  Information saturated (h1) model          Structured
  Standard Errors                             Standard

Latent Variables:
                    Estimate   Std.Err   z-value   P(>|z|)    Std.lv   Std.all
  intercept =~                                                                
    semester_0        1.0000                                  0.2118    0.7170
    semester_1        1.0000                                  0.2118    0.7100
    semester_2        1.0000                                  0.2118    0.6709
    semester_3        1.0000                                  0.2118    0.6132
    semester_4        1.0000                                  0.2118    0.5508
    semester_5        1.0000                                  0.2118    0.4920
  slope =~                                                                    
    semester_0        0.0000                                  0.0000    0.0000
    semester_1        1.0000                                  0.0669    0.2241
    semester_2        2.0000                                  0.1337    0.4235
    semester_3        3.0000                                  0.2006    0.5807
    semester_4        4.0000                                  0.2674    0.6955
    semester_5        5.0000                                  0.3343    0.7764

Covariances:
                    Estimate   Std.Err   z-value   P(>|z|)    Std.lv   Std.all
  intercept ~~                                                                
    slope            -0.0014    0.0016   -0.8337    0.4045   -0.0963   -0.0963

Intercepts:
                    Estimate   Std.Err   z-value   P(>|z|)    Std.lv   Std.all
   .semester_0        0.0000                                  0.0000    0.0000
   .semester_1        0.0000                                  0.0000    0.0000
   .semester_2        0.0000                                  0.0000    0.0000
   .semester_3        0.0000                                  0.0000    0.0000
   .semester_4        0.0000                                  0.0000    0.0000
   .semester_5        0.0000                                  0.0000    0.0000
    intercept         2.5992    0.0183  141.9471    0.0000   12.2724   12.2724
    slope             0.1063    0.0059   18.1113    0.0000    1.5903    1.5903

Variances:
                    Estimate   Std.Err   z-value   P(>|z|)    Std.lv   Std.all
   .smstr_0 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.4859
   .smstr_1 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.4763
   .smstr_2 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.4253
   .smstr_3 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.3554
   .smstr_4 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.2867
   .smstr_5 (resd)    0.0424    0.0021   20.0000    0.0000    0.0424    0.2287
    intrcpt           0.0449    0.0068    6.5992    0.0000    1.0000    1.0000
    slope             0.0045    0.0007    6.3874    0.0000    1.0000    1.0000

The std.all is what we typically will look at.